Variations in the work function of doped single- and few-layer graphene assessed by Kelvin probe force microscopy and density functional theory

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Variations in the work function of doped single- and few-layer graphene assessed by Kelvin probe

force microscopy and density functional theory

D. Ziegler,1,*P. Gava,2J. G¨uttinger,3F. Molitor,3L. Wirtz,4M. Lazzeri,2A. M. Saitta,2A. Stemmer,1 F. Mauri,2and C. Stampfer3,5

1Nanotechnology Group, ETH Zurich, 8092 Zurich, Switzerland 2Universite Paris 6/7, CNRS IPGP, 75015 Paris, France 3Solid State Physics Laboratory, ETH Zurich, 8093 Zurich, Switzerland

4Institute of Electronics, Microelectronics, and Nanotechnology (IEMN), Department ISEN, CNRS-UMR 8520, B.P. 60069,

59652 Villeneuve d’Ascq Cedex, France

5JARA-FIT and II, Institute of Physics, RWTH Aachen University, 52074 Aachen, Germany

(Received 3 January 2011; revised manuscript received 25 April 2011; published 30 June 2011) We present Kelvin probe force microscopy measurements of single- and few-layer graphene resting on SiO2

substrates. We compare the layer thickness dependency of the measured surface potential with ab initio density functional theory calculations of the work function for substrate-doped graphene. The ab initio calculations show that the work function of single- and bilayer graphene is mainly given by a variation of the Fermi energy with respect to the Dirac point energy as a function of doping, and that electrostatic interlayer screening only becomes relevant for thicker multilayer graphene. From the Raman G-line shift and the comparison of the Kelvin probe data with the ab initio calculations, we independently find an interlayer screening length in the order of four to five layers. Furthermore, we describe in-plane variations of the work function, which can be attributed to partial screening of charge impurities in the substrate, and result in a nonuniform charge density in single-layer graphene.

DOI:10.1103/PhysRevB.83.235434 PACS number(s): 73.22.Pr, 74.20.Pq, 71.15.Mb, 63.22.Rc


Carbon nanomaterials and in particular single-layer graphene, a truly two-dimensional (2D) solid with unique electronic1,2 and mechanical properties,3 attract increasing interest, mainly due to possible applications in high mobility nanoelectronics,4–7and sensor technologies.8 Since graphene only consists of surface atoms, a number of new schemes for local doping,9 surface functionalization,10 gas sensing, and atomically thin nanoelectronic circuits11,12become accessible. In this context, the graphene-substrate interaction and, more generally, the interaction between graphene and its environ-ment plays a crucial role as it limits the intrinsic electron mobility in graphene.13,14 The interaction with the substrate and environment is associated with the formation of disorder potential which, in turn, reduces mobilities, limits energy gaps in nanoribbons15,16and bilayer graphene,17and has significant impact on the noise level in graphene devices in general.18 Detailed studies elucidated substrate-induced electron-hole puddles,19 doping domains in graphene,20 and local quan-tum capacitances.21 Kelvin probe microscopy experiments on single- and few-layer graphene have been reported.22,23 Datta et al.22discussed the surface potential as a function of layer thickness and compared their result with Thomas-Fermi theory accounting for interlayer screening of the system. Yu et al.23 have shown that the work function of single-and bilayer graphene can be widely tuned by changing the carrier density. They explain the doping-induced change in the work function using the Fermi level shift obtained from band structure calculations. Here, we report on Kelvin probe force microscopy (KFM) in ambient conditions of single-, bi-, and few-layer graphene resting on SiO2. We compare the KFM data with ab initio density functional theory (DFT)

calculations of the work function for substrate-doped (i.e., surface doped) graphene. The calculations take into account two effects: the substrate-induced charge on the system, and the interlayer screening, i.e., the charge redistribution among the layers. In particular, our calculations show that for single-and bilayer graphene the work function is mainly determined by the Fermi level shift with respect to the Dirac point, i.e., the substrate-induced charge, which confirms the theoretical model used by Yu et al.23 The effect due to the interlayer screening becomes important for multilayer graphene thicker than bilayer. Furthermore, we compare our results with Raman G-line shifts, and discuss substrate-induced variations of the work function, which provide insight into the disorder potential at elevated doping levels in single-layer graphene.

This paper is organized as following: In Sec. II, we describe the experimental setup and present Kelvin probe measurements as well as Raman spectroscopy data. In Sec. III, we introduce the ab initio DFT approach for calculating the work function as function of the number of graphene layers. Finally, we compare the experimental Kelvin probe data with the ab initio DFT calculations in Sec.IV.


Kelvin probe measurements are performed under ambient conditions using a multimode microscope (Veeco) operated by a Nanonis SPM controller (SPECS). First, the topogra-phy of each scan line is recorded using tapping mode for distance control. In a second pass, at a constant distance from the sample (lift height= 5–50 nm), we perform KFM using amplitude modulation (AM-KFM). The frequency of the applied electrical modulation, Vac, is matched to the


-0.5 -1.5 -1 -2 Vac Vdc Vdc (V) Photodiode Laser Cantilever 3.2 nm







Z Kelvin Probe Controller

FIG. 1. (Color online) Schematic illustration of the Kelvin probe force microscopy setup used to investigate graphene flakes resting on SiO2. In the three-dimensional rendering of the topography brighter

areas indicate higher recorded contact potential difference (Vdc) (see

scale bar). A lower Vdc i.e. higher work function is recorded over

areas of single-layer (1LG) than six-layer (6LG) regions.

cancels electrostatic forces by adjusting the tip bias voltage,

Vdc, until zero amplitude is reached.24,25 Here, Vdc

compen-sates for potential differences between tip and sample, and if the tip work function tipis known, the work function of the sample is given by s= tip− eVdc. The local variations of

the work function, however, can be expressed independent of

tipby S=−eVdc. An example of such a measurement

is shown in Fig.1, where the topography is mapped onto the third dimension (z-axis) and the recorded surface potential (Vdc) is color coded. Figures2(a)and 2(c) show topography

and Kelvin probe measurements of an isolated graphene flake consisting of single-layer (1LG) and bilayer (2LG) regions [see labels in Figs.2(a)and 2(c)]. The samples were obtained by mechanical exfoliation of bulk graphite, as described in Refs.4

and26. The Raman data as shown in Fig.2(b)are acquired by using a laser excitation of 2.33 eV through a single-mode optical fiber, whose spot size is limited by diffraction to ∼400 nm. The width and shape of the 2D Raman line [see, e.g., Fig.3(c)] confirm the number of graphene layers26,27and single-layer regions can be well distinguished from bilayer regions in the full width at half maximum (FWHM) of the 2D line as shown in Fig.2(b)(see areas separated by white dotted line).

In Fig.2(d), we show a histogram analysis of the acquired Kelvin probe data over an area enclosing the graphitic flake as indicated in Fig. 2(c) (see black dashed line). Gaussian fitting reveals the average values of the measured surface potentials on the single- and bilayer graphene regions [see labels in Figs. 2(a) and 2(d)], which reproducibly show a difference of Vdc(2−1)≈ 66 mV [Fig. 2(d)]. The strong scattering, i.e., the width of the distribution of the surface potential on the single- and bilayer region of δVdc≈ 25 mV

can most likely be attributed to an inhomogeneously charged background [see, e.g., right edge of Fig.2(c)]. The isolated graphitic flake has been charged by bringing the metallic tip in electrical contact and applying a tip voltage (Vdc) of either 1 V

360 380 400 420 440 460 480 (a) (d) (b) (c) Topography 1LG 2LG 2LG 1LG 1LG 2LG Raman Histogram Kelvin Probe FWHM 2D (e) (f) (V )dc 33cm-1 2 µm Vdc (mV)

Pixel Frequency (arb.units)

240 mV 380 mV 620 mV 950 mV 380 mV 1330 mV 2060 mV 380 mV 2440 mV

FIG. 2. (Color online) (a) AFM topography image of an isolated graphitic flake consisting of single- and bilayer regions (see labels and white dashed line along the boundaries), highlighted and proved by a confocal Raman map (b) showing the FWHM of the 2D Raman line. (c), (e), and (f) Kelvin probe data, i.e., surface potential measurements. (d) Histogram analysis of the acquired Kelvin probe data [taken from the area enclosed by the black dashed line shown in panel (c)]. The two peaks can be attributed to the single- (1LG) and bilayer (2LG) regions and a surface potential difference of Vdc(2−1)≈ 66 mV is observed.

[Fig.2(e)] or 2 V [Fig.2(f)]. Since the highly doped Si substrate has been grounded, this corresponds to an added charge carrier density of roughly n≈ αVdc, where α= 7.2 × 1010 cm−2

V−1 (Ref. 4), leading to n≈ 7.2 × 1010 cm−2 and n 1.4× 1011 cm−2, respectively. In Figs. 2(c), 2(e), and 2(f), the relative color scale is kept constant and just the overall charge dependent offset has been adapted (see color scales). Interestingly, the surface potential difference Vdc(2−1) is—

within the error bars—not affected by the overall charging of the flake, indicating that such small charging and in particular charge rearrangements across single- and bilayer graphene regions do not strongly alter the work function difference

(2−1)S = −eVdc(2−1), which is in agreement with ab initio calculations as shown below.








I (a.u) I (arb.u) 6LG 1LG 3.2 nm I(arb.units) I(arb.units) (a) (b) (c) (d) (f) (e)

FIG. 3. (Color online) (a) 2D-FWHM Raman map of a few-layer graphene flake with sections ranging from six layers down to a single layer (taken before electrically contacting the flake). (b) Raman map of the G-line intensity where (c) and (d) are the full Raman spectra taken on two different region [see labels in (b)]. (e) The AFM topography (with inset indications of the number of layers) and (f) Kelvin probe image of the same flake after contacting [image corresponds to the dashed square in panel (a)]. The labeled boxes therein relate to Fig.6.

in Fig. 3(b)]. Figure3(c), for example, shows a single-layer spectrum with a narrow 2D line (and no D line), whereas Fig.3(d)shows a Raman spectrum of a six-layer-thick region confirmed by atomic force microscope (AFM) measurements. Topography and Kelvin probe measurements of this very same flake are shown in Figs. 3(e) and 3(f). The different layer dependent surface potential values Vdccan be clearly identified

in Fig.3(f)[see also numbered labels in Fig.3(e)]. We observe that the surface potential Vdc increases with increasing layer

thickness, which predicts p-doping of the sample.28 For the surface potential difference between the single- and bilayer region we find Vdc(2−1)≈ 68 meV, which is very close to the

value reported above. However, other groups reported larger values of Vdc(2−1) (Refs. 22 and 23), but as shown in the

following, this can be attributed to different doping, which strongly depends on the preparation of the sample.29,30 The three closeups of Figs.3(e)and 3(f) (see white labeled 1.5 by 1.5 μm boxes therein) will be discussed in Fig.6.

The observed differences in work function compared to a multilayer area (N )S are depicted in Fig.4 (full squares).

Compared to bulk graphite we observe a difference of 150, 82, and 26 meV, for single-, bi-, and three-layer graphene, respectively. The particularly large scattering of (2)S may be related to the relatively narrow bilayer region. The inset in Fig.4shows the measured Raman G-line shift as a function of number of layers. The doping n of the sample can be determined from the G-line position ωGusing the calculated

relations n (ωG) for single-layer31and bilayer graphene.32The

G-line shift is nearly symmetric and positive for electron

Number of layers N ΔΦ 1 2 3 4 5 6 1582 1583 1584 1585 1586 1587 1588 1589 nil -G p t ei sn oo i ) -1 m c( Number of layers N Exp. Raman G-line

Exp. Kelvin Probe x10 cm

FIG. 4. (Color online) Work function differences as a function of the layer thickness (full squares), extracted by histogram analysis of the acquired Kelvin probe data. (Inset) Raman G-line shift. Triangles indicate ab initio DFT calculations of the work functions for a doping level of n =−4.5× 1012cm−2.

and hole doping, therefore the sign of the doping cannot be extracted from the measured G-line shift. However, assuming a hole doping on this flake, we find a doping level of n≈ −(2.5 ± 1)× 1012cm−2for the single-layer and n≈ −(7.5 ± 1.5)×1012cm−2for the bilayer. Similar values can be obtained using the experimentally measured ωGas a function of n.20,33,34

These Raman measurements suggest that single- and bilayer graphene on the same sample have different doping values, which will be further discussed below. For a number of layers larger than two, the analogous relation n (ωG) has not been

determined yet.

III. DENSITY FUNCTIONAL THEORY CALCULATION In order to compare the Kelvin probe measurements with

ab initio DFT calculations, we used the PWscf code of


charge. Positive n corresponds to electron doping and negative

ncorresponds to hole doping. The surface charge density leads to a discontinuity of the electric field across the graphene layers equal to σ/0, where 0 is the permittivity of the vacuum. We assume that the counter charge density is located on the bottom side of the system (e.g., in the gate electrode) such that the electric field is E= σ/0 below the graphene sheet and, assuming absence of adsorbates, we set E= 0 above the flake. The electric field configuration is reproduced in our calculations by introducing monopole and dipole potentials in the edges of the super cell as described by Gava et al.38 The resulting potential energy in the direction perpendicular to the sheet is shown in Fig. 5(a) for single- and bilayer graphene (a planar average in the directions parallel to the sheets has been performed). The electric field is obtained from the derivative of the planar average, calculated in the region outside few-layer graphene where the potential energy is linear and E uniform. The zero electric field condition on the top side of the system is required to define the work function S,

which is the minimum energy for an electron to escape into vacuum.  is the potential energy at infinite distance from the system [in Fig.5(a), it is set to zero for convenience]. The work function Sof multilayer graphene is then defined as

S= − F, (1)

where Fis the Fermi level.

In the insets of Fig. 5(a), we show the calculated band structure for p-doped single- and bilayer graphene, around the K point in the Brillouin zone (BZ). The effect of the electric field on the system is twofold: (i) the band structure is distorted and the alignment of the Dirac point is modified with respect to the vacuum energy and (ii) the Fermi level is shifted with respect to the Dirac point because of removal or addition of charges. For the single layer, the electric field does not affect the linear crossing of the π bands (which is dictated by the hexagonal symmetry of the system). However, for the bilayer the zero-field electric band structure displays a parabolic crossing of the π bands at K,39and this crossing is lifted in presence of a finite electric field.

In Fig.5(b), we show the doping dependence of the work function Sof single and few-layer graphene, calculated with

Eq. (1), which includes both the effect of the Fermi level shift due to the induced charge and the effect due to the different screening properties of multilayer graphene. The work function for single-layer graphene shows a strong doping dependence, while its variation in n decreases with increasing number of layers. This is due to the fact that the doping charge (per unit area) can distribute over different layers. Thus, the more layers there are, the lower is the Fermi level shift.

The doping dependence of S can be due to both, the

variation of the Fermi level with respect to the Dirac point d,

or an electric-field-induced variation of d with respect to the

vacuum level:

S= (− d)+ (d− F), (2)

where d is calculated as the average energy of the π bands

at K.40 For zero doping, our DFT calculations show that (− d) does not depend on the number of layers and it is

Planar a



age of potential ener

gy (eV ) Layer 1 Layer 2 cm ΦS cm εd εF

FIG. 5. (Color online) (a) Planar average of the potential en-ergy across 1LG (dashed line) and 2LG (continuous line), for a doping value n= −4.5 × 1012cm−2, calculated in DFT. (Insets)

Corresponding band structures of 1LG and 2LG around the K point, illustrating the difference in work function 1LG

S for single- and 2LGS

for bilayer graphene. The planar average and band structures are plotted with respect to the energy in the vacuum ∞, here set to zero for convenience. (b) Ab initio DFT calculated work function, S, as a

function of doping concentration n, for multilayers graphene obtained from Eq. (1). The vertical dashed line indicates a doping value of n= −4.5 × 1012cm−2for which we found the best agreement with the


≈4.23 eV. Assuming that (− d) is independent of the


S≈ (d − F)+ 4.23 eV. (3)

The comparison between Figs.5(b)and5(c)[i.e., Eqs. (2) and (3)] shows that the approximated expression [Eq. (3)] well reproduces the doping dependence of the work function for single-layer graphene. For the largest values of n considered in Fig. 5(c), the error in the work function difference with respect to n= 0 is around 4%. Equation (3) also reasonably reproduces the doping dependence of the work function for bilayer graphene, even if in this case the error is increased with respect to single-layer, being around 30%. In single- and bilayer graphene, the work function is thus mainly determined by the Fermi level shift with respect to the Dirac point as a function of the charge density, justifying the analysis of Yu

et al.23 Our findings also explain andependence of 

S on

the charge density for the single layer, and a linear dependence on n for the double layer. Indeed, for the single layer

d− F = −sign(n)


|n|, (4)

and for the biayer

d− F = − π¯h2n


, (5)

where vF is the Fermi velocity and m1 is the effective mass. Moreover, in our DFT calculations charges are adsorbed from the bottom of the system. However, in the case of single- and bilayer graphene, we have shown that the major effect which determines the work function variation is the Fermi level shift with respect to the Dirac point, i.e., the total doping charge

n, while the Dirac point shift with respect to the vacuum is negligible. Therefore, from this decomposition, we conclude that in the case of single- and bilayer graphene the work function results to be independent of whether such charge is adsorbed from the top or the bottom of the system. For more than two layers, the error in the work function difference with respect to zero doping (n= 0) becomes larger than in bilayer graphene when using Eq. (3) and, therefore, the work function cannot be interpreted as a simple shift in the Fermi energy. In particular, one can notice that for a large number of layers the work function variation with n is negligible [Fig. 5(b)], because the graphene layers screen the electric field on the bottom side of the sample. This screening is neglected in Eq. (3) and therefore, for the comparison with experimental results, we have used the work function values as calculated in Eq. (1) and shown in Fig.5(b).

IV. COMPARISON AND DISCUSSION A. Layer dependent work function differences

In Fig.4, we compare the experimentally observed work function differences as a function of the layer thickness with the results of our ab initio DFT calculations. The experimental results are in reasonable agreement with the ab

initio calculations when assuming an overall doping value of n= −4.5×1012cm−2(see triangles and continuous line). We observe that the work function difference decreases signifi-cantly for the first few-layers but it is no longer layer-dependent

for systems with more than four to five layers. This is in good agreement with recent transport experiments on double-gated few-layer graphene systems, where an interlayer screening length of 1.2± 0.2 nm has been reported.41In contrast to the Kelvin probe data, we observe for the layer-dependent Raman G-line position (see inset in Fig.4) a slight monotonic change even for larger layer numbers N. This is mainly attributed to the fact that interlayer screening does not play an important role for the Raman measurement, since all layers are contributing to the Raman signal. This is in contrast to the Kelvin probe measurement, where the surface potential is playing the crucial role and lower lying layers are partially screened. However for very few-layer graphene samples (one to three layers), where the doping-induced shift of the charge neutrality point is the dominating mechanism for shifting the work function difference (see discussion in Sec. III), the Raman data and the Kelvin probe data are in reasonably good agreement. For example, we observe an enhancement of the doping level of the bilayer section in both data sets. The origin of this elevated doping value is not known but it may be related to the relatively small bilayer graphene area on the investigated sample, where edge effects may play an important role, may also explain the rather extended error bar. However, a systematic increased doping level for bilayer graphene cannot be excluded, since it could be in agreement with recent experimental results on chemically functionalized graphene where it has been found that adsorbates bind differently to single- and bilayer graphene.42

From Fig.5(b), we can also notice that at the estimated p-doping values (indicated by the vertical line), the small charging n= 1.4 ×1011cm−2 induced upon contact with the biased AFM tip only leads to a minor modification of the work functions, i.e., of Vdc(2−1)as observed in our experiment and mentioned above.

B. Lateral variations in the work function of single- and few-layer graphene


20 mV 20 mV 1.7 nm

1.7 nm

pixel frequency (arb.units.) pixel frequency (arb.units.)

pixel frequency (arb.units.)



Topography Surface Potential Histograms

Surface Potential Fluctuations (mV)

20 mV 1.7 nm (a) (b) (c) SiO 2 (d) (e) (f) (g) (h) (i) −100 −5 0 5 10 50 100 150 200 −10 −5 0 5 10 0 20 40 60 80 −10 −5 0 5 10 0 20 40 60 80

FIG. 6. (Color online) Closeup views of topography (1.5 × 1.5 μm) and corresponding Kelvin probe data recorded on SiO2

(a) and (d), 1LG (b) and (e), and 6LG (c) and (f) as indicated in Figs.3(e)and 3(f). The topography images exhibit the presence of contaminations, but do not show significant differences depending on the layer thickness. The Kelvin probe measurements, in contrast, show striking differences. Whereas the work function measured on 6LG is very homogeneous, we see significant variations on 1LG, showing that different substrate doping cannot be sufficiently screened. The width of the distribution δVdc(1)≈ 9 mV can be attributed to such local doping variations resulting in a charge inhomogeneity, while δVdc(6)≈

3.6 mV lies in the range of the noise level of our KFM measurement setup. The variations are indicated by the corresponding histograms shown in panels (g) and (h).

partial screening of charge impurities that induce a doping. The corresponding histograms [Figs. 6(g) and 6(h)] show that single-layer graphene partially screens the impurities as expressed by a narrowing of the distribution in Fig.6(h). For single-layer graphene, we observe an asymmetric distribution of the surface potential as highlighted by the two arrows in Fig.6(h), whereas Figs.6(g)and6(i)exhibit a rather symmetric distribution. This asymmetry reflects the overall p-doping of the sample resulting in an excess of holes in the graphene

layer, which allows more efficient screening of electrons. Additionally, the linear density of states suppresses the screening of more positive charge impurities in the p-doped regime [see right arrow in Fig.6(h)]. Interestingly, we observe these strong variations in doping at elevated doping levels which consequently leads to the conclusion that significant disorder is also present at rather high carrier densities.


In summary, we presented Kelvin probe measurements on single-, bi-, and few-layer graphene flakes resting on SiO2. The Kelvin probe measurements revealed significant work function variations as a function of the number of graphene layers, in quantitative agreement with ab initio DFT computed work functions for p-doped multilayer graphene. Moreover, our calculations show that for single- and bilayer graphene the work function variation with doping is mainly due to the shift of the Fermi energy with respect to the Dirac point, while for more than two layers the effects due to interlayer screening become relevant and have to be carefully taken into account. Measured and computed work function independently lead to an interlayer screening length in the order of four to five layers. The measurements, moreover, indicate different doping values for single- and bilayer graphene on the same sample. These findings are consistent with our Raman G-line shift mea-surements. We observed significant variations of the surface potential on 1LG. These variations are related to the partial screening of charge impurities in the substrate which results in a nonuniform charge density in graphene, and presently determine one of the major limitations in the performance of state-of-the-art graphene based electronic devices.


The authors wish to thank D. Bishop, P. Brouwer, K. Ensslin, F. Guinea, T. Ihn, and R. W. Stark for helpful discussions and C. Hierold for providing access to the Raman spectrometer. Support by the ETH FIRST Lab and financial support by the Swiss National Science Foundation and NCCR Nanoscience are gratefully acknowledged. Calculations were performed at the IDRIS supercomputing center (Project No. 096128).

*Present address: Lawrence Berkeley National Laboratory, Molecular

Foundry, 1 Cyclotron Road, Berkeley, California 94720, USA;

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